Intrinsic compound kernel estimates for the transition probability density of Lévy-type processes and their applications

INTRINSIC COMPOUND KERNEL ESTIMATES FOR THE TRANSITION PROBABILITY DENSITY OF LÉVY-TYPE PROCESSES AND THEIR APPLICATIONSStarting with an integro-differential operator L, C2∞ ℜn, we prove that its C∞ℜn-closure is the generator of a Feller process X, which admits a transition probability density. To construct this transition probability density, we develop a version of the parametrix method and a verification procedure, which proves that the constructed object is the claimed one. As a part of the construction, we prove the intrinsic upper and lower estimates on the density. As an application of the constructed estimates we state the necessary and separately sufficient conditions under which a given Borel measure belongs to the Kato and Dynkin classes with respect to the constructed transition probability density.

Numerical methods for Stochastic differential equations: two examples

The goal of this paper is to present a series of recent contributions arising in numerical probability. First we present a contribution to a recently introduced problem: stochastic differential equations with constraints in law, investigated through various theoretical and numerical viewpoints. Such a problem may appear as an extension of the famous Skorokhod problem. Then a generic method to approximate in a weak way the invariant distribution of an ergodic Feller process by a Langevin Monte Carlo simulation. It is an extension of a method originally developed for diffusions and based on the weighted empirical measure of an Euler scheme with decreasing step. Finally, we mention without details a recent development of a multilevel Langevin Monte Carlo simulation method for this type of problem.

Feller evolution systems: Generators and approximation

A time and space inhomogeneous Markov process is a Feller evolution process, if the corresponding evolution system on the continuous functions vanishing at infinity is strongly continuous. We discuss generators of such systems and show that under mild conditions on the generators a Feller evolution can be approximated by Markov chains with Lévy increments. The result is based on the approximation of the time homogeneous spacetime process corresponding to a Feller evolution process. In particular, we show that a d-dimensional Feller evolution corresponds to a (d + 1)-dimensional Feller process. It is remarkable that, in general, this Feller process has a generator with discontinuous symbol.

APPROXIMATION OF FELLER PROCESSES BY MARKOV CHAINS WITH LÉVY INCREMENTS

We consider Feller processes whose generators have the test functions as an operator core. In this case, the generator is a pseudo differential operator with negative definite symbol q(x, ξ). If |q(x, ξ)| < c(1 + |ξ|2), the corresponding Feller process can be approximated by Markov chains whose steps are increments of Lévy processes. This approximation can easily be used for a simulation of the sample path of a Feller process. Further, we provide conditions in terms of the symbol for the transition operators of the Markov chains to be Feller. This gives rise to a sequence of Feller processes approximating the given Feller process.